scholarly journals A UNIFIED TREATMENT OF CONVEXITY OF RELATIVE ENTROPY AND RELATED TRACE FUNCTIONS, WITH CONDITIONS FOR EQUALITY

2010 ◽  
Vol 22 (09) ◽  
pp. 1099-1121 ◽  
Author(s):  
ANNA JENČOVÁ ◽  
MARY BETH RUSKAI
Keyword(s):  
Relative Entropy ◽  
Matrix Inequalities ◽  
Strong Subadditivity ◽  
Special Cases ◽  
Joint Convexity

We consider a generalization of relative entropy derived from the Wigner–Yanase–Dyson entropy and give a simple, self-contained proof that it is convex. Moreover, special cases yield the joint convexity of relative entropy, and for Tr K* Ap K B1-p Lieb's joint concavity in (A, B) for 0 < p < 1 and Ando's joint convexity for 1 < p ≤ 2. This approach allows us to obtain conditions for equality in these cases, as well as conditions for equality in a number of inequalities which follow from them. These include the monotonicity under partial traces, and some Minkowski type matrix inequalities proved by Carlen and Lieb for [Formula: see text]. In all cases, the equality conditions are independent of p; for extensions to three spaces they are identical to the conditions for equality in the strong subadditivity of relative entropy.

scholarly journals On quantum quasi-relative entropy

2019 ◽  
Vol 31 (07) ◽  
pp. 1950022
Author(s):  
Anna Vershynina
Keyword(s):  
Relative Entropy ◽  
Logarithmic Function ◽  
Operator Inequalities ◽  
Operator Convex Function ◽  
Strong Subadditivity ◽  
Entropy Formula ◽  
Quantum Relative Entropy ◽  
Skew Information ◽  
Explicit Bounds ◽  
Joint Convexity

We consider a quantum quasi-relative entropy [Formula: see text] for an operator [Formula: see text] and an operator convex function [Formula: see text]. We show how to obtain the error bounds for the monotonicity and joint convexity inequalities from the recent results for the [Formula: see text]-divergences (i.e. [Formula: see text]). We also provide an error term for a class of operator inequalities, that generalizes operator strong subadditivity inequality. We apply those results to demonstrate explicit bounds for the logarithmic function, that leads to the quantum relative entropy, and the power function, which gives, in particular, a Wigner–Yanase–Dyson skew information. In particular, we provide the remainder terms for the strong subadditivity inequality, operator strong subadditivity inequality, WYD-type inequalities, and the Cauchy–Schwartz inequality.

Monotonicity of quantum relative entropy and recoverability

2015 ◽  
Vol 15 (15&16) ◽  
pp. 1333-1354 ◽  
Author(s):  
Mario Berta ◽  
Marius Lemm ◽  
Mark M. Wilde
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Relative Entropy ◽  
Remainder Term ◽  
Conditional Entropy ◽  
Quantum Information Theory ◽  
Partial Trace ◽  
Quantum Operations ◽  
Strong Subadditivity ◽  
Joint Convexity

The relative entropy is a principal measure of distinguishability in quantum information theory, with its most important property being that it is non-increasing with respect to noisy quantum operations. Here, we establish a remainder term for this inequality that quantifies how well one can recover from a loss of information by employing a rotated Petz recovery map. The main approach for proving this refinement is to combine the methods of [Fawzi and Renner, 2014] with the notion of a relative typical subspace from [Bjelakovic and Siegmund-Schultze, 2003]. Our paper constitutes partial progress towards a remainder term which features just the Petz recovery map (not a rotated Petz map), a conjecture which would have many consequences in quantum information theory. A well known result states that the monotonicity of relative entropy with respect to quantum operations is equivalent to each of the following inequalities: strong subadditivity of entropy, concavity of conditional entropy, joint convexity of relative entropy, and monotonicity of relative entropy with respect to partial trace. We show that this equivalence holds true for refinements of all these inequalities in terms of the Petz recovery map. So either all of these refinements are true or all are false.

scholarly journals Approximate Tensorization of the Relative Entropy for Noncommuting Conditional Expectations

2021 ◽  
Author(s):  
Ivan Bardet ◽  
Ángela Capel ◽  
Cambyse Rouzé
Keyword(s):  
Relative Entropy ◽  
Von Neumann Algebras ◽  
Logarithmic Sobolev Inequality ◽  
Spin Systems ◽  
Von Neumann ◽  
Strong Subadditivity ◽  
Finite Dimensional ◽  
Lattice Spin ◽  
Lattice Spin Systems ◽  
Conditional Expectations

AbstractIn this paper, we derive a new generalisation of the strong subadditivity of the entropy to the setting of general conditional expectations onto arbitrary finite-dimensional von Neumann algebras. This generalisation, referred to as approximate tensorization of the relative entropy, consists in a lower bound for the sum of relative entropies between a given density and its respective projections onto two intersecting von Neumann algebras in terms of the relative entropy between the same density and its projection onto an algebra in the intersection, up to multiplicative and additive constants. In particular, our inequality reduces to the so-called quasi-factorization of the entropy for commuting algebras, which is a key step in modern proofs of the logarithmic Sobolev inequality for classical lattice spin systems. We also provide estimates on the constants in terms of conditions of clustering of correlations in the setting of quantum lattice spin systems. Along the way, we show the equivalence between conditional expectations arising from Petz recovery maps and those of general Davies semigroups.

BEYOND STRONG SUBADDITIVITY? IMPROVED BOUNDS ON THE CONTRACTION OF GENERALIZED RELATIVE ENTROPY

1994 ◽  
Vol 06 (05a) ◽  
pp. 1147-1161 ◽  
Author(s):  
MARY BETH RUSKAI
Keyword(s):  
Relative Entropy ◽  
Discrete Case ◽  
Sobolev Inequalities ◽  
Completely Positive ◽  
Logarithmic Sobolev Inequalities ◽  
Strong Subadditivity

New bounds are given on the contraction of certain generalized forms of the relative entropy of two positive semi-definite operators under completely positive mappings. In addition, several conjectures are presented, one of which would give a strengthening of strong subadditivity. As an application of these bounds in the classical discrete case, a new proof of 2-point logarithmic Sobolev inequalities is presented in an Appendix.

scholarly journals Entropic Steering Criteria: Applications to Bipartite and Tripartite Systems

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 763 ◽  
Author(s):  
Ana Costa ◽  
Roope Uola ◽  
Otfried Gühne
Keyword(s):  
Relative Entropy ◽  
Uncertainty Relation ◽  
Uncertainty Relations ◽  
W States ◽  
Entropic Uncertainty Relations ◽  
Entropic Uncertainty Relation ◽  
Local Measurements ◽  
Action At A Distance ◽  
Quantum Steering ◽  
Joint Convexity

The effect of quantum steering describes a possible action at a distance via local measurements. Whereas many attempts on characterizing steerability have been pursued, answering the question as to whether a given state is steerable or not remains a difficult task. Here, we investigate the applicability of a recently proposed method for building steering criteria from generalized entropic uncertainty relations. This method works for any entropy which satisfy the properties of (i) (pseudo-) additivity for independent distributions; (ii) state independent entropic uncertainty relation (EUR); and (iii) joint convexity of a corresponding relative entropy. Our study extends the former analysis to Tsallis and Rényi entropies on bipartite and tripartite systems. As examples, we investigate the steerability of the three-qubit GHZ and W states.

scholarly journals Geometric measures of discordlike quantum correlations based on Tsallis relative entropy

2019 ◽  
Vol 17 (04) ◽  
pp. 1950034
Author(s):  
Weijing Li
Keyword(s):  
Relative Entropy ◽  
Quantum Correlation ◽  
Quantum Correlations ◽  
Partial Coherence ◽  
Good Measure ◽  
Coherence Measure ◽  
Geometric Measure ◽  
Special Cases ◽  
Analytic Expression ◽  
Correlation Measures

A kind of new geometric measure of quantum correlations is formulated. The proposed formulation is in terms of the quantum Tsallis relative entropy and can naturally be viewed as a one-parameter extension quantum discordlike measure that satisfies all requirements of a good measure of quantum correlations. It is of an elegant analytic expression and contains several existing good quantum correlation measures as special cases. The partial coherence measure is also investigated.

scholarly journals Synchronization for an Array of Coupled Cohen-Grossberg Neural Networks with Time-Varying Delay

2011 ◽  
Vol 2011 ◽  
pp. 1-22 ◽  
Author(s):  
Haitao Zhang ◽  
Tao Li ◽  
Shumin Fei
Keyword(s):  
Neural Networks ◽  
Convex Combination ◽  
Network Models ◽  
Linear Matrix ◽  
Matrix Inequalities ◽  
Neural Network Models ◽  
Time Varying Delay ◽  
Weighting Matrix ◽  
Special Cases ◽  
Varying Delay

This paper makes some great attempts to investigate the global exponential synchronization for arrays of coupled delayed Cohen-Grossberg neural networks with both delayed coupling and one single delayed one. By resorting to free-weighting matrix and Kronecker product techniques, two novel synchronization criteria via linear matrix inequalities (LMIs) are presented based on convex combination, in which these conditions are heavily dependent on the bounds of both the delay and its derivative. Owing to that the addressed system can include some famous neural network models as the special cases, the proposed methods can extend and improve those earlier reported ones. The efficiency and applicability of the presented conditions can be demonstrated by two numerical examples with simulations.

scholarly journals Pairwise quantum and classical correlations in multi-qubits states via linear relative entropy

2014 ◽  
Vol 12 (06) ◽  
pp. 1450035 ◽  
Author(s):  
M. Daoud ◽  
R. Ahl Laamara ◽  
H. El Hadfi
Keyword(s):  
Relative Entropy ◽  
Qubit State ◽  
Mixed States ◽  
Special Cases ◽  
Pairwise Correlations ◽  
Classical Correlations ◽  
Additivity Relation ◽  
Quantum And Classical Correlations ◽  
Explicit Expressions

The pairwise correlations in a multi-qubit state are quantified through a linear variant of relative entropy. In particular, we derive the explicit expressions of total, quantum and classical bipartite correlations. Two different bi-partioning schemes are considered. We discuss the derivation of closest product, quantum–classical and quantum–classical product states. We also investigate the additivity relation between the various pairwise correlations existing in pure and mixed states. As illustration, some special cases are examined.

MASTER–SLAVE SYNCHRONIZATION OF LUR'E SYSTEMS WITH GENERAL SECTOR-BOUNDED NONLINEARITIES

2009 ◽  
Vol 19 (02) ◽  
pp. 517-529 ◽  
Author(s):  
QING-LONG HAN ◽  
DRISS MEHDI ◽  
DONGSHENG HAN
Keyword(s):  
Time Delay ◽  
Linear Matrix Inequalities ◽  
Lyapunov Functional ◽  
Linear Matrix ◽  
Matrix Inequalities ◽  
Lur’E Systems ◽  
Sector Condition ◽  
Special Cases ◽  
Delay Feedback Control ◽  
Delay Dependent

This paper deals with master–slave synchronization for Lur'e systems subject to a more general sector condition by using time delay feedback control. A new Lyapunov–Krasovskii functional and a new Lur'e–Postnikov Lyapunov functional are proposed to obtain some new delay-dependent synchronization criteria, which are formulated in the form of linear matrix inequalities (LMIs). These criteria cover some existing results as their special cases. An example shows that the result derived in this paper significantly improves some existing ones.

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